# Monday, March 13

:::{.definition title="Hopf algebras"}
Define a Hopf algebra as

- $A\in \kalg$ with $\pi: A\tensorpowerk 2 \to A$ multiplication and unit $\eta: k\to A$,
- $\Delta: A\to A\tensorpowerk 2$ comultiplication
- $\eps: A\to k$ counit/augmentation 
- $s: A\to A^\op$ the coinverse/antipode
- Coassociativity: $(\Delta\tensor \id)\Delta = (1\tensor \Delta) \Delta$
- $(\eps \tensor \id) \Delta = \id = (\id\tensor \eps) \Delta$.
- $\pi (s \tensor \id) \Delta = \eta \eps = \pi(\id \tensor s) \Delta$.

This can be summarized in a diagram: 

\begin{tikzcd}
	& {A\tensorpowerk 2} && {A\tensorpowerk 2} \\
	A && k && A \\
	& {A\tensorpowerk 2} && {A\tensorpowerk 2}
	\arrow["\Delta", from=2-1, to=1-2]
	\arrow["{\id\tensor s}", from=1-2, to=1-4]
	\arrow["\Delta"', from=2-1, to=3-2]
	\arrow["\eta"', from=2-1, to=2-3]
	\arrow["{s\tensor \id}"', from=3-2, to=3-4]
	\arrow["\pi", from=1-4, to=2-5]
	\arrow["\varepsilon"', from=2-3, to=2-5]
	\arrow["\pi"', from=3-4, to=2-5]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMCwxLCJBIl0sWzEsMCwiQVxcdGVuc29ycG93ZXJrIDIiXSxbMywwLCJBXFx0ZW5zb3Jwb3dlcmsgMiJdLFsxLDIsIkFcXHRlbnNvcnBvd2VyayAyIl0sWzMsMiwiQVxcdGVuc29ycG93ZXJrIDIiXSxbMiwxLCJrIl0sWzQsMSwiQSJdLFswLDEsIlxcRGVsdGEiXSxbMSwyLCJcXGlkXFx0ZW5zb3IgcyJdLFswLDMsIlxcRGVsdGEiLDJdLFswLDUsIlxcZXRhIiwyXSxbMyw0LCJzXFx0ZW5zb3IgXFxpZCIsMl0sWzIsNiwiXFxwaSJdLFs1LDYsIlxcdmFyZXBzaWxvbiIsMl0sWzQsNiwiXFxwaSIsMl1d)

**Cocommutative** if $z\Delta = \Delta$ where $z: A\tensorpowerk 2\selfmap$ is the twist map $z(a\tensor b) = b\tensor a$.

:::

:::{.example title="?"}
1. For $G\in \Fin\Grp$, the group algebra $kG \da \ts{\sum_{g\in G} a_g g \st a_g\in k }$ with multiplication $\qty{\sum a_g g}\qty{\sum b_h h} \da \sum_{g,h} a_g b_h gh$, $\Delta(g) = g\tensor g, \eps(g) = 1_k, s(g) = g\inv$ is a commutative Hopf algebra.

2. For $V\in \mods{k}$, the tensor algebra $T(V)$ is a cocommutative Hopf algebra with $\Delta(x) \da x\tensor 1 + 1\tensor x, \eps(x) = 0, s(x) = -x$.

3. Similarly $\Sym(V)$ is commutative and cocommutative, and $\Extalg V$ is cocommutative.

4. For $\lieg\in \liealg\slice k$, the enveloping algebra $\mcu(\lieg) = T(V)/ (v\tensor w-w\tensor v = [vw])$ is cocommutative.

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:::{.remark}
Sweedler's Hopf algebra: over $\characteristic k \neq 2$, let $A\da T_2\in \kalg$ be presented as $\gens{g,x\st g^2=1, x^2=0, xg=-gx}$.
This is a Hopf algebra under $\Delta(g)\da g\tensor g, \Delta(x) \da 1\tensor x + x\tensor g, \eps(g) = 1, \eps(x) = 0, s(g) = g, s(x) = -xg$.
Note $\dim_k T_2 = 4$, generated by $\gens{1,g,x,xg}_k \in \kmod$.
This is the smallest example of a non-commutative *and* non-cocommutative Hopf algebra.
How to prove this: 
\[
(\Delta\tensor 1) \Delta(g) &= (\Delta\tensor 1)(g\tensor g) = g\tensorpower{}{3} \\
(1 \tensor \Delta) \Delta(g) &= ( \Delta\tensor 1)(1 \tensor x + x \tensor g) = (1 \tensor \Delta)(g \tensor g) = g\tensor{}{3} \\
(\Delta\tensor 1) \Delta(x) &= \cdots 1 \tensor 1 \tensor x + 1 \tensor x \tensor g + x \tensor g \tensor g \\
(1 \tensor \Delta) \Delta(x) &= \cdots = \text{same} \\
(\eps \otimes 1) \Delta(g) = (\eps \otimes 1)(g \otimes g) = 1 \otimes g \cong g = \id_A(g) \\
(1 \otimes \eps) \Delta(g) = (1 \otimes \eps)(g \otimes g) = g \otimes 1 \cong g = \id_A(g) \\
(\eps \otimes 1) \Delta(x) &= \cdots = 1 \otimes x + 0 \otimes g \cong x = \id_A(x)
(1 \otimes \eps) \Delta(x) = \cdots = 1 \otimes 0 + x \otimes 1 \cong x = \id_A(x)
,\]
etc.
Non-cocommutativity follows from checking the following:
\[
z \Delta(x) = z(1 \otimes x + x \otimes g) = x \otimes 1 + g \otimes x \neq \Delta(x)
.\]
Later we'll see it has a Gerstenhaber bracket, but is not quasitriangular.
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:::{.remark}
Taft algebras: for $n\geq 2, q = \zeta_n \in k$, let $A\da T_n \da \gens{g,x \st g^n = 1, x^n=0, xg = qgx}$ with \( \Delta(g) = g \otimes g, \Delta(x) = 1 \otimes x + x \otimes g, \eps(g) = 1, \eps(x) = 0, s(g) = g\inv, s(x) = -x g\inv \).
Then $A$ is Hopf, with $\dim A = n^2$, non-commutative and non-cocommutative.
:::